Rigidity of $\varepsilon$-harmonic maps of low degree

Abstract

. In 1981, Sacks and Uhlenbeck introduced their famous α -energy as a way to approxi-mate the Dirichlet energy and produce harmonic maps from surfaces into Riemannian manifolds. However, the second and third authors together with Malchiodi ([11], [12]) showed that for maps between two-spheres this method does not capture every harmonic map. They established a gap theorem for α -harmonic maps of degree zero and also showed that below a certain energy bound α -harmonic maps of degree one are rotations. We establish similar results for ε -harmonic maps u ε : S 2 → S 2 , which are critical points of the ε -energy introduced by the second author in [9]. In particular, we similarly show that ε -harmonic maps of degree zero with energy below 8 π are constant and that maps of degree ± 1 with energy below 12 π are of the form Rx with R ∈ O (3). Moreover, we construct non-trivial ε -harmonic maps of degree zero with energy > 8 π .